We use the concept of unimodular random graph to show that the branching simple random walk on $\mathbb{Z}^{d}$ indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if $d \leq 4$.

We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this…

Recently, Abraham and Delmas constructed the distributions of super-critical Lévy trees truncated at a fixed height by connecting super-critical Lévy trees to (sub)critical Lévy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this…